Fiber bundles with category morphisms as fibersYour question is missing many details. In what sense are the elements of your fibres morphisms? What are your "transfer functions" supposed to achieve? Why are you suddenly able to differentiate when you were not able to before? etc.

Minimum number of sets required for a good open coverHere is a crude lower bound for the minimum: by considering the homology of the simplicial complex associated with a good open cover, we see that a non-zero homology group in dimension $n$ implies that the good open cover has at least $n + 1$ elements; but the homology of the simplicial complex associated with a good open cover is isomorphic to the homology of the space itself (when the space is nice enough), so this gives a lower bound for all good open covers.

On two definitions of the nerve of a simplicial categoryFirst things first: simplicial sets considered as objects in the Joyal model category must not be confused with simplicial sets considered as objects in the Kan–Quillen model category. The "geometric realisation" of a bisimplicial set is all about the latter, whereas the homotopy coherent nerve is all about the former. Secondly, ordinary categories are too much of a special case to tell you anything interesting here; instead you should be looking at, say, a simplicial monoid considered as a one-object simplicial category.